

int(rmutil)                                  R Documentation

_N_u_m_e_r_i_c_a_l _I_n_t_e_g_r_a_t_i_o_n _o_f _a _F_u_n_c_t_i_o_n

_D_e_s_c_r_i_p_t_i_o_n_:

     `int' performs numerical integration of a given func-
     tion using either Romberg integration or algorithm 614
     of the collected algorithms from ACM. Only the former
     is vectorized. The latter appeared in ACM-Trans. Math.
     Software, Vol.10, No. 2, Jun., 1984, p. 152-160 and
     uses formulae optimal in certain Hardy spaces h(p,d);
     see Sikorski,K., Optimal quadrature algorithms in HP
     spaces, Num. Math., 39, 405-410 (1982).

     Functions may have singularities at one or both end-
     points of the interval (a,b).

_U_s_a_g_e_:

     int(f, a="-infty", b="infty", type="Romberg", eps=1.0e-6, max, d, p=0)

_A_r_g_u_m_e_n_t_s_:

       f: The function (of one variable) to integrate,
          returning either a scalar or a vector.

       a: A scalar or vector giving the lower bound. If non-
          numeric, taken to be -infty, in which case it must
          be the same for the whole vector.

       b: A scalar or vector giving the upper bound. If non-
          numeric, taken to be infty, in which case it must
          be the same for the whole vector.

    type: The algorithm to be used, by default Romberg inte-
          gration.  Otherwise, it uses the TOMS614 algo-
          rithm.

     eps: Precision.

     max: For Romberg, the maximum number of steps, by
          default set to 16. For TOMS614, the maximum number
          of function evaluations, by default set to 100.

       d: For Romberg, the number of extrapolation points so
          that 2k is the order of integration, by default
          set to 5; d=2 is Simpson's rule. For TOMS614,
          heuristic termination = any real number; determin-
          istic termination = a number in the range 0 < d <
          pi/2 by default, set to 1.

       p: For TOMS614, p = 0: heuristic termination, p = 1:
          deterministic termination with the infinity norm,
          p > 1: deterministic termination with the p-th
          norm.

_A_u_t_h_o_r_(_s_)_:

     J.K. Lindsey

_E_x_a_m_p_l_e_s_:

     f <- function(x) sin(x)+cos(x)-x^2
     int(f, a=0, b=2)
     #
     f <- function(x) exp(-(x-2)^2/2)/sqrt(2*pi)
     int(f, a=0:3)
     1-pnorm(0:3, 2)

